Published by the Department of Computer Science, The University of Chicago.

Abstract

A simplicial complex is d-collapsible if it can be reduced to an empty complex by repeatedly removing (collapsing) a face of dimension at most $d-1$ that is contained in a unique maximal face. We prove that the algorithmic question whether a given simplicial complex is $d$-collapsible is NP-complete for $d \ge 4$ and polynomial time solvable for $d\le 2$.

As an intermediate step, we prove that d-collapsibility can be recognized by the greedy algorithm for $d \le 2$, but the greedy algorithm does not work for $d \ge3$.

A simplicial complex is d-representable if it is the nerve of a collection of convex sets in ${\mathbb R}^d$. The main motivation for studying $d$-collapsible complexes is that every $d$-representable complex is $d$-collapsible. We also observe that known results imply that $d$-representability is NP-hard to decide for $d \ge 2$.

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Submitted October 23, 2008, revised June 15, 2010, published June 21, 2010.

© 2010 Martin Tancer
Licensed under a Creative Commons Attribution License

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